OF CONCIIOLOGY. 229 



at the Troy uicetiiifr, Sept., 1870, (of which a synopsis Avas pub- 

 lished in the Am. Naturalist, Nov., 1870, p. 561), I restricted 

 the whole order within its present limits from the researches 

 above mentioned. Among the fruits of these investigations was 

 the definite exclusion of the G-adiniidce from the order, (see Am. 

 Jour. Conch, vi. p. 8, 1870). It is proper to state that, on 

 general considerations, Prof. Theodore Gill, in his unpublished 

 manuscripts, had adopted similar limits for the order, though the 

 conclusions to which I have been led were the results of entirely 

 independent investigations. 



A brief synopsis of the order and its subdivisions as here 

 adopted, was, for the sake of insuring priority, presented to the 

 Boston Society of Natural History in September, 1870, and pub- 

 lished in advance of the Proceedings, February 7th, 1871. 



Formulatini/ the Dentition. 



Attempts have frequently been made, with greater or less 

 success, to devise a method of formulating, mathematically, the 

 different styles of dentition. Having used several methods, and 

 finding them all more or less deficient in some particulars, I 

 would propose the following as an improvement upon any now 

 in use, and likely to meet any demands that can reasonably be 

 made upon it. For greater compactness the formula may be 

 written in the form of a fraction, with the rhachidian tooth as a 

 numerator ; the lateral teeth enclosed in brackets and the uncini 

 outside of the brackets, as a denominator. The median line in 

 the latter maybe indicated by a period and the teeth of the 

 lateral and uncinal series severally connected by a plus sign. 

 Each separate tooth may be denoted by a fraction, the denomi- 

 nator representing the number of cusps, and if there is more than 

 one tooth similarly formed, the numerator of the fraction will 

 serve to indicate the number. For instance, suppose we have 

 a radula with a median tooth possessing five cusps, this tooth 

 would be indicated by a fraction thus ; — I. If there were six 

 inner laterals each with three cusps they would be formulated 

 by the fraction, | ; another lateral tooth outside of them with 

 one cusp or entirely simple would be written j, and connect- 

 ed with the other six with a plus sign. Ten simple uncini on 

 ea/ch side would form a fraction -j^', and the whole formula 



would be i 



LOd-f^.l+D'y". The absence of teeth in either series 



might be indicated by a cipher, thus q 



i_0M4-6 6_|_i)ij) or I 



0.0. 



16 



